Regular Languages for Contracting Geodesics

Eike, Joshua; Zalloum, Abdul

doi:10.48550/arxiv.1809.02692

Cited by 2 publications

(3 citation statements)

References 7 publications

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“…Theorem D provides an extension of a result by Eike and Zalloum who showed that strongly contracting geodesics with a fixed parameter form a regular language in any finitely generated group [EZ18]. Strongly contracting geodesics are a stronger form of hyperbolic-like behavior in a group compared to Morse geodesics.…”

Section: Introductionsupporting

confidence: 54%

“…We first prove uq = uwa is a geodesic word in Cay(G, A). This part of the proof closely follows [EZ18] and the cone types section of [BH09]. Recall, for ω ∈ A ⋆ , ℓ(ω) denotes the word length in the free monoid A ⋆ , while |ω| denotes the length of a geodesic word in Cay(G, A) that represents the group element ω.…”

Section: Theorem 39 Let G Be a Finitely Generated Group With Finite G...mentioning

confidence: 88%

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Regularity of Morse geodesics and growth of stable subgroups

Cordes¹,

Russell²,

Davide³

et al. 2020

Preprint

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We prove that Morse local-to-global groups grow exponentially faster than their infinite index stable subgroups. This generalizes a result of Dahmani, Futer, and Wise in the context of quasi-convex subgroups of hyperbolic groups to a broad class of groups that contains the mapping class group, CAT(0) groups, and the fundamental groups of closed 3-manifolds.To accomplish this, we develop a theory of automatic structures on Morse geodesics in Morse local-to-global groups. Other applications of these automatic structures include a description of stable subgroups in terms of regular languages, rationality of the growth of stable subgroups, density in the Morse boundary of the attracting fixed points of Morse elements, and containment of the Morse boundary inside the limit set of any infinite normal subgroup.

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Section: Introductionsupporting

confidence: 54%

Section: Theorem 39 Let G Be a Finitely Generated Group With Finite G...mentioning

confidence: 88%

Regularity of Morse geodesics and growth of stable subgroups

Cordes¹,

Russell²,

Davide³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…A classical application of the local-to-global property of quasi-geodesic in hyperbolic groups is Cannon's proof that the geodesics of a hyperbolic group form a regular language [Can84]. In [EZ18], the local nature of Morse geodesics in CATp0q spaces is used to show that the M -Morse geodesics of a CATp0q group also form a regular language. Extending these result to all Morse local-to-global groups would produce new results for both the mapping class group and 3-manifold groups.…”

Section: A Long Standing Open Question Of Farb and Mosher Makes Combi...mentioning

confidence: 99%

The local-to-global property for Morse quasi-geodesics

Russell¹,

Davide²,

Tran³

2019

Preprint

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We show the mapping class group, CATp0q groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasigeodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan-Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

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Regular Languages for Contracting Geodesics

Cited by 2 publications

References 7 publications

Regularity of Morse geodesics and growth of stable subgroups

Regularity of Morse geodesics and growth of stable subgroups

The local-to-global property for Morse quasi-geodesics

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